Spatial and temporal scaling affected by system inhomogeneity: atomic, microscopic and macroscopic.
G. C. Siha b *
a School of Mechanical Engineering, East China University of Science and Technology, Shanghai 200237, China
b Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem PA 18015, USA
*Email: gcs@ecust.edu.cn Fax: +86 (21) 6425-3500.
Extended Abstract
For the better part of the 20th century, the thought was that sufficient test data would provide information to understand how material micro-structure effects could be used to predict macroscopic bulk behavior. The expectation, however, were short lived. The use of non-linear continuum mechanics was equally disappointing. The impetus of nanotechnology since the 1950s has finally settled on the prospect of science and technology for the 21st century. Microelectronic and microbiology headed the top of the shopping list. The survival of some of the more traditional fields such as continuum mechanics and material science had to justify their existence by showing their relevance with reference to the current trend of atomic or sub-atomic simulations. While the physicists and chemists were more at ease with the current trend, many of the engineers were treading unfamiliar grounds. In the areas of material engineering and continuum mechanics, the field of “Mesomechanics” [1] attracted attention owing to the necessity of scaling shifting in size and time [2, 3].
When devices were made smaller and smaller, not only quantification of material properties must be re-considered in terms of theoretic interpretation, but the basic fundamentals of continuum mechanics required scrutinizing. This is simply because many of the empirical laws established from rigid body concepts will not hold when the surface-to-volume ratio becomes large. Keep in mind that classical continuum mechanics requires that the surface-to-volume ratio to vanish in the limit such that only the bulk properties of the continuum need to be considered. This is equivalent to assuming that the bulk properties of the uniaxial test can be used locally for a system where the stress and/or strain states are not uniform. By tradition, invokement of homogeneity was taken for granted, an implication that has not been over looked for too long [4].The uniaxial curve in general is assumed to undergo a series of equilibrium states although this depends on the strain rates. Such a restriction is often violated in tests when the specimen size falls outside of the requirements of ASTM and/or the ASME codes. Not until recently, the design philosophy has been to use small specimen data for larger structural components. The reverse becomes problematic because of the violation of homogeneity when specimen size is reduced.
The challenge, therefore, is to be able to address the integrity of submicron size device parts that may not follow the traditional criteria of failure by fracture of structural components or bulk material damage.
Modeling of multiscale material damage theories raises several basic issues. To begin with, conditions must be invoked to connect the results observed at the different temporal and spatial scales. Up to now, discussions seem to be confined to a very narrow range of size and time. Furthermore, the models seldom address the effect of the initial or residual state as differentiated from the performance sate. The former becomes increasingly more important as the size of device is reduced to sub-microns. This is the rule rather than the exceptions in microelectronics. It is in this nanoelectronics region that the electron transport behavior does not strictly obey quantum mechanics nor classical physics. It has been referred to as the mesoscopic electronics region, particularly with reference to power dissipation. Micro-chips should be kept sufficiently cool so that they will operate in a stable manner. And yet the density of the transistors must also be high and closely packed. The optimum balance can be achieved only by knowing the limits of how effectively a very small device can dissipate heat. At the mesoscopic scale, the non-equilibrium isoenergy density approach [5] can be applied. It has solved many problems with mesoscopic phenomena [6]. The non-equilibrium theory [5] stresses in particular the power dissipation that is derived directly by considering the mutual interaction of mechanical and thermal effects without invoking artificial dissipation laws and/or constitutive relations. The dissipation has been shown to depend sensitively on the temporal and spatial characters of the local deformation. Under extension, a cooling period has been observed [6] that precedes heating for solid, liquid and gas. This fundamental feature is not considered in classical physics. It can be very important for the design of microchips in electronics.
The objective of this work is directed towards the development of physical models that can relate results at different scale ranges with account for change of system homogeneity as the region of interest is reduced in size. A corresponding increase in the time scale follows automatically. In order to preserve the use of equilibrium mechanics in the ranges referred to as atomic, microscopic and macroscopic, attention will be focused in the region where damage is concentrated in the form of a singularity for the stress and energy density fields. The displacement field is required to remain finite and continuous even though its cyclic value may become multi-valued. Cross scale transition is made possible by imposing scale invariant criterion based on the “force” and/or “energy” quantities. In passing, the Cauchy-Born rule [7,8] in crystal elasticity should not be applied freely without careful considerations. This is because the rule invokes the use of the scale sensitive strain energy density function to cross scale. The singularity representation approach [9] will be first applied to illustrate how disorders in the system at the microscopic and atomic scales can interact. The former and latter will be associated, respectively, with micro-cracking and dislocations. Non-linear equation are solved for the coupling of the micro-energy and dislocation-energy density functions, designated by Wmicro and W disln in normalized form, respectively. They will be used to derive the length of inhomogeneity 
disln and
micro for the system. The connection between the microscopic and atomic scale ranges is complete only when the stresses, displacements and energy densities satisfy the scale invariant conditions by application of the force and/or energy (not energy density) criteria. The formulation entails several orders of magnitude extending from 10-11 to 10-1 on the lineal scale. The same can be done between the microscopic and macroscopic scale with inhomegeneity length of
micro and 
macro.while Wmicro and Wmacro are the corresponding normalized volume energy density functions. It follows that the connection among the atomic, microscopic and macroscopic scales can then be made by following the same procedure although the details are by no means straight forward. The scheme can be repeated to extend the range of scale shifting. The time rate enters into the problem from the hypothesis that [10]
The time rate of the energy density factor remains unchanged
between two successive scale ranges in size and/or time.
Symbolically, the above statement when written for the atomic and microscopic scale takes the form
where dot represents time derivative. The factor 
is unity if the curve for versus distance r is a perfect hyperbola. The above relation was obtained by using the relation =[11] with r being a length parameter. When 
is not unity, scale shifting is accompanied by a change in the homogeneity of the system. The foregoing transitions are necessitated by the absence of a theory that can carry results from the atomic to the macroscopic via the microscopic scale range. The same bottle neck is present in physics where gravity and quantum principle do not meet. This appears to be equivalent to crossing the forbidden mesoscopic zone through which reconciliation of the particulate and the continuum seems to take place. Such a situation occurs over and over again in physics. Explanation of the modern light theory relies on the dualism of infinitesimal particle-like objects and yet at the same time suggest that light exhibits all the features of a continuous wave. Until the clouds associated with these fundamental issues are cleared, the piece-meal approach of mesomechanics may have to prevail in the mean time.
References
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[7] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, xford, 1954.
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